Monotonic instability and overstability in two-dimensional electrothermohydrodynamic flow

Electrothermohydrodynamic convection driven by strong unipolar charge injection in the presence of a stabilizing inverse thermal gradient between two parallel electrodes is investigated by a linear stability analysis and a numerical simulation. The generalized Schur decomposition is used to solve for the eigenvalues of the linearized system revealing the critical parameters. The two relaxation time lattice Boltzmann method coupled to a fast Poisson solver is used to resolve the nonlinear system for the spatiotemporal distribution of flow field, electric field, charge density, and temperature. With strong charge injection and high electric Rayleigh number, the system exhibits electrothermoconvective vortices. The interactions between the stabilizing buoyancy force and the destabilizing electric force lead to overstability, where the flow constantly oscillates when instability evolves. A two-stage bifurcation is observed for overstability near the threshold Rayleigh number with a significant change in phase and amplitude. The effects of ion mobility and thermal diffusivity are characterized by the ratio of the counteracting forces.

Figure 1

Physical model and boundary conditions of ETHD convection in a player layer of dielectric liquid. The external forces (Coulomb and buoyancy force) interact with viscous and inertial forces of the dielectric fluid, inducing convection after an initial perturbation.

Figure 2

Hydrostatic base state of the TRT LBM and Fast Poisson solver, unified SRT LBM [12], and the analytical solution [94, 95] for $C=0.1$ and $C=10$, $Fe=4000$. (a) Electric field and (b) charge density.

Figure 3

Temperature (a) and charge-density (b) field of ETHD convection system heated from above. $C=M=Pr=10$, $Ra=977.6$, and $T=230$ (monotonic mode). The computational domain is doubled periodically for visualization.

Figure 4

Evolution of maximum $u_y$ of monotonic (solid line) and oscillatory (dashed line) modes after initial perturbation at $C=Pr=M=10$, $Fe=2000$, $Ra=977.6$, and $T=225$.

Figure 5

Neutral stability curve ($T_c$ vs wave number $\alpha$) for monotonic modes (MM) and oscillatory modes (OM) at $Pr=M=C=10$ and $Fe=2000$.

Figure 6

Stability threshold $T_c$ is proportional to Ra for ETHD system heat from above. The critical value of Ra ($R{a}_c$) at $C=Pr=M=10$ and $Fe=2000$ marks the transition from monotonic instability to overstability.

Figure 7

Growth rate of the maximum $u_y$ increases linearly with respect to T. $T_c$ is approximated by linear extrapolation. $Pr=M=C=10$ and $Fe=2000$.

Figure 8

Electric Nusselt number Ne vs T for $Ra=977.6$, $Pr=M=C=10$, and $Fe=2000$ for (a) monotonic mode and (b) oscillatory mode. The bifurcation with respect to the monotonic mode is subcritical with a hysteresis loop, while the one for the oscillatory mode is supercritical. Overstability exhibits two stages of bifurcation. Panel (i) amplifies the first stage where the frequency remains unchanged. Panels (ii) and (iii) show the Ne evolution with time at the first and second stage, respectively. Linear stability analysis predicts the first stage of bifurcation. Error bars represent one standard deviation of the oscillation.

Figure 9

Snapshots of (a) temperature, (b) charge density, and (c) vorticity field for $T=225$, $Ra=977.6$, $Pr=M=C=10$, and $Fe=2000$. The oscillatory mode is at the first stage of bifurcation, where the wavelength remains constant. (Full video is provided in the Supplemental Material. [96]).

Figure 10

Snapshots of (a) temperature, (b) charge density, and (c) vorticity field for $T=245$, $Ra=977.6$, $Pr=M=C=10$, and $Fe=2000$. The oscillatory mode is at the second stage of bifurcation, where multiple wavelengths coexist. (Full video is provided in the Supplemental Material. [96]).

Figure 11

Monotonic instability threshold $T_c$ remains the same with varying Pr and M. Overstability threshold $T_c$ decreases with increasing Pr or decreasing M.

Figure 12

Electric Nusselt number Ne vs T for monotonic modes at $Ra=1000$, $C=10$, and $Fe=2000$. $T_f$ increases for a decreasing Pr and an increasing M.

Figure 13

Time evolution of maximum wall normal velocity $u_y$ for varying M at $Ra=1000$, $Pr=C=10$, and $Fe=2000$. The value is normalized by the $u_{\mathrm{drift}}$ at $M=10$ for consistency. The embedded panels show the temporal structures of maximum $u_y$ for each M.

Figure 14

Time evolution of maximum wall normal velocity $u_y$ for varying Pr at $Ra=1000$, $M=C=10$, and $Fe=2000$. The value is normalized by the $u_{\mathrm{drift}}$ at $Pr=10$ for consistency. The embedded panels show the temporal structures of maximum $u_y$ for each Pr.